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<div class="doctools">
<h1 class="doctools_title">math::calculus::symdiff(n) 1.0.1 tcllib &quot;Symbolic differentiation for Tcl&quot;</h1>
<div id="name" class="doctools_section"><h2><a name="name">Name</a></h2>
<p>math::calculus::symdiff - Symbolic differentiation for Tcl</p>
</div>
<div id="toc" class="doctools_section"><h2><a name="toc">Table Of Contents</a></h2>
<ul class="doctools_toc">
<li class="doctools_section"><a href="#toc">Table Of Contents</a></li>
<li class="doctools_section"><a href="#synopsis">Synopsis</a></li>
<li class="doctools_section"><a href="#section1">Description</a></li>
<li class="doctools_section"><a href="#section2">Procedures</a></li>
<li class="doctools_section"><a href="#section3">Expressions</a></li>
<li class="doctools_section"><a href="#section4">Examples</a></li>
<li class="doctools_section"><a href="#section5">Bugs, Ideas, Feedback</a></li>
<li class="doctools_section"><a href="#see-also">See Also</a></li>
<li class="doctools_section"><a href="#copyright">Copyright</a></li>
</ul>
</div>
<div id="synopsis" class="doctools_section"><h2><a name="synopsis">Synopsis</a></h2>
<div class="doctools_synopsis">
<ul class="doctools_requirements">
<li>package require <b class="pkgname">Tcl 8.5</b></li>
<li>package require <b class="pkgname">grammar::aycock 1.0</b></li>
<li>package require <b class="pkgname">math::calculus::symdiff 1.0.1</b></li>
</ul>
<ul class="doctools_syntax">
<li><a href="#1"><b class="cmd">math::calculus::symdiff::symdiff</b> <i class="arg">expression</i> <i class="arg">variable</i></a></li>
<li><a href="#2"><b class="cmd">math::calculus::jacobian</b> <i class="arg">variableDict</i></a></li>
</ul>
</div>
</div>
<div id="section1" class="doctools_section"><h2><a name="section1">Description</a></h2>
<p>The <b class="cmd">math::calculus::symdiff</b> package provides a symbolic differentiation
facility for Tcl math expressions. It is useful for providing derivatives
to packages that either require the Jacobian of a set of functions or else
are more efficient or stable when the Jacobian is provided.</p>
</div>
<div id="section2" class="doctools_section"><h2><a name="section2">Procedures</a></h2>
<p>The <b class="cmd">math::calculus::symdiff</b> package exports the two procedures:</p>
<dl class="doctools_definitions">
<dt><a name="1"><b class="cmd">math::calculus::symdiff::symdiff</b> <i class="arg">expression</i> <i class="arg">variable</i></a></dt>
<dd><p>Differentiates the given <i class="arg">expression</i> with respect to the specified
<i class="arg">variable</i>. (See <span class="sectref"><a href="#section3">Expressions</a></span> below for a discussion of the
subset of Tcl math expressions that are acceptable to
<b class="cmd">math::calculus::symdiff</b>.)
The result is a Tcl expression that evaluates the derivative. Returns an
error if <i class="arg">expression</i> is not a well-formed expression or is not
differentiable.</p></dd>
<dt><a name="2"><b class="cmd">math::calculus::jacobian</b> <i class="arg">variableDict</i></a></dt>
<dd><p>Computes the Jacobian of a system of equations.
The system is given by the dictionary <i class="arg">variableDict</i>, whose keys
are the names of variables in the system, and whose values are Tcl expressions
giving the values of those variables. (See <span class="sectref"><a href="#section3">Expressions</a></span> below
for a discussion of the subset of Tcl math expressions that are acceptable
to <b class="cmd">math::calculus::symdiff</b>. The result is a list of lists:
the i'th element of the j'th sublist is the partial derivative of
the i'th variable with respect to the j'th variable. Returns an error if
any of the expressions cannot be differentiated, or if <i class="arg">variableDict</i>
is not a well-formed dictionary.</p></dd>
</dl>
</div>
<div id="section3" class="doctools_section"><h2><a name="section3">Expressions</a></h2>
<p>The <b class="cmd">math::calculus::symdiff</b> package accepts only a small subset of the expressions
that are acceptable to Tcl commands such as <b class="cmd">expr</b> or <b class="cmd">if</b>.
Specifically, the only constructs accepted are:</p>
<ul class="doctools_itemized">
<li><p>Floating-point constants such as <b class="const">5</b> or <b class="const">3.14159e+00</b>.</p></li>
<li><p>References to Tcl variable using $-substitution. The variable names
must consist of alphanumerics and underscores: the <b class="const">${...}</b> notation
is not accepted.</p></li>
<li><p>Parentheses.</p></li>
<li><p>The <b class="const">+</b>, <b class="const">-</b>, <b class="const">*</b>, <b class="const">/</b>. and <b class="const">**</b>
operators.</p></li>
<li><p>Calls to the functions <b class="cmd">acos</b>, <b class="cmd">asin</b>, <b class="cmd">atan</b>,
<b class="cmd">atan2</b>, <b class="cmd">cos</b>, <b class="cmd">cosh</b>, <b class="cmd">exp</b>, <b class="cmd">hypot</b>, <b class="cmd"><a href="../log/log.html">log</a></b>,
<b class="cmd">log10</b>, <b class="cmd">pow</b>, <b class="cmd">sin</b>, <b class="cmd">sinh</b>. <b class="cmd">sqrt</b>, <b class="cmd">tan</b>,
and <b class="cmd">tanh</b>.</p></li>
</ul>
<p>Command substitution, backslash substitution, and argument expansion are
not accepted.</p>
</div>
<div id="section4" class="doctools_section"><h2><a name="section4">Examples</a></h2>
<pre class="doctools_example">
math::calculus::symdiff::symdiff {($a*$x+$b)*($c*$x+$d)} x
==&gt; (($c * (($a * $x) + $b)) + ($a * (($c * $x) + $d)))
math::calculus::symdiff::jacobian {x {$a * $x + $b * $y}
                         y {$c * $x + $d * $y}}
==&gt; {{$a} {$b}} {{$c} {$d}}
</pre>
</div>
<div id="section5" class="doctools_section"><h2><a name="section5">Bugs, Ideas, Feedback</a></h2>
<p>This document, and the package it describes, will undoubtedly contain
bugs and other problems.
Please report such in the category <em>math :: calculus</em> of the
<a href="http://core.tcl.tk/tcllib/reportlist">Tcllib Trackers</a>.
Please also report any ideas for enhancements you may have for either
package and/or documentation.</p>
<p>When proposing code changes, please provide <em>unified diffs</em>,
i.e the output of <b class="const">diff -u</b>.</p>
<p>Note further that <em>attachments</em> are strongly preferred over
inlined patches. Attachments can be made by going to the <b class="const">Edit</b>
form of the ticket immediately after its creation, and then using the
left-most button in the secondary navigation bar.</p>
</div>
<div id="see-also" class="doctools_section"><h2><a name="see-also">See Also</a></h2>
<p><a href="calculus.html">math::calculus</a>, <a href="interpolate.html">math::interpolate</a></p>
</div>
<div id="copyright" class="doctools_section"><h2><a name="copyright">Copyright</a></h2>
<p>Copyright &copy; 2010 by Kevin B. Kenny &lt;kennykb@acm.org&gt;<br>
Redistribution permitted under the terms of the Open Publication License &lt;http://www.opencontent.org/openpub/&gt;</p>
</div>
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